Question

Integrate the function:

`1/(xsqrt(ax - x^2)) ["Hint : Put x" = a/t]`

Answer 1

Let `1/(xsqrt(ax - x^2))`

Put `x = a/t` 

dx = `- a/t^2 dt`

Now, `xsqrt(ax - x^2) = a/tsqrt(a xx a/t - a^2/t^2)`

`= a^2/t sqrt(1/t - 1/t^2) = a^2/t^2 sqrt(t - 1)`

`therefore I = 1/(a^2/t^2 sqrt(t - 1)) xx (- a)/t^2  dt`

`= - 1/a int 1/sqrt(t - 1)  dt`

`= - 1/a ((t - 1)^(- 1/2 + 1))/(- 1/2 + 1) + C`

`= - 1/a (t - 1)^(1/2)/(1/2) + C`

`= - 2/a sqrt(t - 1) + C`

`= - 2/a sqrt(a/x - 1) + C`

`= - 2/a sqrt((a - x)/x) + C`